In highly scattering regimes, the transport equation with anisotropic boundary conditions has a limit in which the leading behavior of its solution is determined by the solution of a diffusion equation with associated boundary conditions. In order for a numerical scheme to be effective in these regimes, it must have both a correct interior diffusion limit and a correct boundary condition limit. The behavior of several numerical methods are studied in these limits and formulas for the resulting diffusion equations and its boundary conditions are derived. Theoretic and numerical results show that with correct diffusion limits, the numerical methods will give promising results with coarse grids throughout the domain, even if the boundary layers are not resolved. We also prove that with correct diffusion limits, the numerical solutions will converge to the transport solution uniformly in ε, although the collision operators have a ε⁻¹ contribution to the truncation error that generally gives rise to a nonuniform consistency with the transport equation for small ε. In last part of this dissertation we study numerical methods for the hyperbolic systems with long time parabolic behavior. In this regime the lower order terms of the hyperbolic systems break the conservation law and the systems become parabolic. Most of the numerical methods for conservation laws fail to capture this long time behavior, as shown in our analysis. We will solve the general Riemann problem of the shallow water equations and use it to modified higher order Godunov schemes in order to capture the long time behavior of the nonlinear river equations.

In highly scattering regimes, the transport equation with anisotropic boundary conditions has a limit in which the leading behavior of its solution is determined by the solution of a diffusion equation with associated boundary conditions. In order for a numerical scheme to be effective in these regimes, it must have both a correct interior diffusion limit and a correct boundary condition limit. The behavior of several numerical methods are studied in these limits and formulas for the resulting diffusion equations and its boundary conditions are derived. Theoretic and numerical results show that with correct diffusion limits, the numerical methods will give promising results with coarse grids throughout the domain, even if the boundary layers are not resolved. We also prove that with correct diffusion limits, the numerical solutions will converge to the transport solution uniformly in ε, although the collision operators have a ε⁻¹ contribution to the truncation error that generally gives rise to a nonuniform consistency with the transport equation for small ε. In last part of this dissertation we study numerical methods for the hyperbolic systems with long time parabolic behavior. In this regime the lower order terms of the hyperbolic systems break the conservation law and the systems become parabolic. Most of the numerical methods for conservation laws fail to capture this long time behavior, as shown in our analysis. We will solve the general Riemann problem of the shallow water equations and use it to modified higher order Godunov schemes in order to capture the long time behavior of the nonlinear river equations.

en_US

dc.type

text

en_US

dc.type

Dissertation-Reproduction (electronic)

en_US

dc.subject

Dissertations, Academic

en_US

dc.subject

Civil engineering

en_US

thesis.degree.name

Ph.D.

en_US

thesis.degree.level

doctoral

en_US

thesis.degree.discipline

Applied Mathematics

en_US

thesis.degree.discipline

Graduate College

en_US

thesis.degree.grantor

University of Arizona

en_US

dc.contributor.advisor

Levermore, C. David

en_US

dc.contributor.committeemember

Hyman, Mac

en_US

dc.contributor.committeemember

Brio, Moysey

en_US

dc.contributor.committeemember

Bayly, Bruce J.

en_US

dc.identifier.proquest

9210295

en_US

dc.identifier.oclc

712064839

en_US

All Items in UA Campus Repository are protected by copyright, with all rights reserved, unless otherwise indicated.